**R E S E A R C H**

**Open Access**

## A positive solution of a

*p*

## -Laplace-like

## equation with critical growth

### Zhou Yang

1*_{, Di Geng}

1_{and Huiwen Yan}

2
*_{Correspondence:}

yangzhou1975@yahoo.com.cn
1_{School of Mathematics Science,}
South China Normal University,
Guangzhou, 510631, China
Full list of author information is
available at the end of the article

**Abstract**

The existence of a positive solution of a*p*-Laplace-like equation with critical growth is
established by the generalization to the concentration-compactness principle and
the Sobolev inequality under some proper assumptions. Moreover, we achieve some
regularity results of the solution.

**MSC:** 35J65

**Keywords:** *p*-Laplace-like operator; critical growth; concentration-compactness

principle; weakly continuity

**1 Introduction**

This paper is devoted to the existence of a positive solution of the following*p-Laplace-like*
problem with critical growth:

⎧ ⎨ ⎩

–div(a(∇u)) =*f*(x,*u),* *x*∈,

*u*= , *x*∈*∂*, (.)

whereis a smooth bounded domain in**R***N*_{, <}_{p}_{<}_{N, and the functions}_{a,}_{f}_{satisfy some}

proper conditions, the details of which are described later.

There were many papers about the existence of the solution of*p-Laplacian problems*
involving critical growth such as [–]. In them,*a(ξ*) =|*ξ*|*p*–_{ξ}_{and}_{f}_{are some concrete}
functions with critical growth, which means that*f*(x,*u)|u|*–*p** _{does not converge to zero}

as*u*→ ∞, where*p**is the critical exponent, *i.e.,p**=*Np/(N*–*p). The *
concentration-compactness principle, which was built by Lions in [, ], plays an important role in
achiev-ing the existence of a nontrivial solution of the problems in them.

The authors proved the existence of a nonnegative and nontrivial solution for a
Dirich-let problem for*p-mean curvature operate with critical growth in [], wherea(ξ*) = ( +
|*ξ*|)*p*– _{ξ}_{,}* _{p}*≥

_{, and}

_{f}_{is some concrete function involving a critical exponent. Since the}

function*a*has an explicit form, the authors can use the concentration-compactness
prin-ciple to achieve their results, too. But if*a*is an abstract function in problem (.), then
the problem becomes more complicated and interesting, and Lions’ C-C principle cannot
be directly applied to it. Thanks to the generalization of the C-C principle in [], we can
establish the existence of a nonnegative and nontrivial solution of equation (.) if we
im-pose some proper conditions on the functions*a*and*f* and make more careful estimates.
Moreover, we achieve some regularity result of the solution and prove the solution is
posi-tive under some proper assumptions. The results can be easily extended to a more general

*p-Laplace-like equation with critical growth and singular weights by the *
Caﬀarelli-Kohn-Nirenberg inequality and the method in [].

Recently, there have been some articles on stochastic partial diﬀerential equations
(SPDEs) involving*p-Laplace operator; see [, ]. Some estimates and properties of the*
solution of the corresponding elliptic equations are important to the research on SPDEs.
So, the results in this paper may be useful in the study of*p-Laplace SPDEs with critical*
growth.

In this paper, we suppose that the potential*a*:**R***N*_{→}_{R}*N*_{satisﬁes the following }

assump-tions.

Let*A*=*A(ξ*) :**R***N*→**R**be of continuous derivative with respect to*ξ* with*a*=∇*ξA*and

satisfy the following conditions:
(A) *A() = ,*

(A) there are*p*> and*m*∈[,*p], three positive constantsa*,*a*and*a*such that

*a*|*ξ*|*p*≤*A(ξ*)≤*a*|*ξ*|*p*+*a*|*ξ*|*m* for all*ξ*∈**R***N*,

(A) *A(ξ*)is strictly convex in*ξ*, that is,A(*ξ*+*η*) <*A(ξ*) +*A(η*)for any*ξ*=*η*∈**R***N*_{,}

(A) there exists a positive number*a*such thatlim|*ξ*|→+∞*a(ξ*)·*ξ*|*ξ*|–*p*=*a*.

We impose some assumptions on the critical nonlinear term*f*(x,*u) :*×**R**→**R**, which
is continuous, as follows:

(B) *f*(x, ) = ,

(B) there is a function*b(x)*∈*L*∞()such that

lim sup

*u→*
sup

*x∈*

*f*(x,*u)u*|u|–*p*≤*b(x),*

(B) there are two positive numbers*c*and*c*such that

lim inf

*u→∞* *x∈*inf

*f*(x,*u)u*|u|–*p**≥*c*, lim sup

*u→∞* sup*x∈*

*f*(x,*u)u*|u|–*p**≤*c*,

(B) denote*F(x,u) =*_{}*uf*(x,*s)ds, which satisﬁes*

lim inf

*u→∞* *x∈*inf

*f*(x,*u)u*–*p***F(x,u)*|*u*|–*p**≥.

Moreover, we suppose*a*and*f* satisfy the next correlation.
(C) there exists a*β*> such that

*pA(*∇*υ*) –*b(x)*|*υ*|*pdx*≥*β*

*A(*∇*υ*)*dx* for any*υ*∈*X,*

where*X*is*W*_{},*p*(),*i.e.*, the completion of*C*_{}∞()with the norm
*u* *X*= (

|∇*u*|

*p _{dx)}*/

*p*

_{.}

in another paper. Since*a(z) is not an explicit function, we need to use the generalization*
of the C-C principle in [] and more subtle estimates to study problem (.).

It is clear that the solution of problem (.) is the critical point of the variational func-tional

*I(u) =*

*A(*∇*u)dx*–

*F(x,u)dx.* (.)

Moreover,*I(u) is continuous diﬀerentiable inX, and its Fréchet derivation is*

*I*(u),*υ*=

*a(∇u)*· ∇*υdx*–

*f*(x,*u)υdx,* ∀*υ*∈*X.* (.)

The ﬁrst main result in this paper is

**Theorem .** *Suppose problem*(.)*satisﬁes assumptions*(A)-(A), (B)-(B)*and*(C).

*Moreover,there exists a nontrivialυ*∈*X such thatυ*≥*and*

(C) sup_{t≥}_{}*I(tυ*) < (* _{p}*–

**)(pS)*

_{p}*p**
*p**–*p _{c}*

*p*
*p*–*p**

,*whereS*=inf*υ*∈X\{} A(∇*υ*) *υ* –* _{p}p**,

*with*

*υ* *qq*=

|*υ*|*qdx*(q≥).

*Then problem*(.)*has a nonnegative and nontrivial solution.*

Since the condition (C) is diﬃcult to check, we give another easily checked theorem.

**Theorem .** *Assume conditions*(A)-(A), (B)-(B)*and*(C)*are satisﬁed and*

(A) *pA(ξ*)≥*a*|*ξ*|*pfor anyξ*∈**R***N*,

(B) *there exists a nonempty setW*⊂*such thatF(x,u) –c*

*p**|*u*|*p*
*

≥*for any*

*x*∈*W*,*u*∈**R**,
(C) lim*ε*→+(*ε*

(*N*–*p*)*m*

(*p*–)*p*–*N*_{+}_{K}

(*φ*))(K(*ψ*))–= ,*whereK*(*ψ*) =

*ε*

(r)*dr,*
*K*(*φ*) =

*ε*

(r)*drwith*

(r) =*ψ*

*ε*–
+*rp*/(*p*–)

*N*–*p*

*p*

*rN*–, *ψ*(r) = min

*u≥r*,*x∈W*

*F(x,u) –c*
*p**|u|

*p**

,

(r) =*φ*

*ε*–

+*rp*/(*p*–)

*N*
*p*

*r*/(*p*–)

*rN*–, *φ*(r) = max
≤|*ξ*|≤r

*pA(ξ*) –*a*|*ξ*|*p*

.

*Then problem*(.)*has a nonnegative and nontrivial solution in X.*

To establish the regularity of the solution*u*and prove*u*> in, we need to impose
stronger assumptions on the potential*a*and the nonlinear term*f*, which are as follows:

(D) *A(ξ*)∈*C*(**R***N*\ {})with*a() =*****and there exist positive numbers*c*and*C,*

*κ*∈[, ]such that for any*ξ*∈**R***N*_{\ {}}_{and}_{η}_{∈}_{R}*N*_{,}

*N*

*i*,*j*=

*∂**A*

*∂ξi∂ξjηiηj*≥*c*

*κ*+|*ξ*|p–|*η*|,

*N*

*i*,*j*=

*∂**A*

*∂ξi∂ξj*

≤*Cκ*+|*ξ*|p–,

**Theorem .** *Assume the assumptions in Theorem*.*or those in Theorem*.*hold,then*
*there exists a constantα*∈(, )*such that the solution u*∈*C*,*α*_{(}_{}_{)}_{if assumption}_{(D)}_{holds.}

*Moreover,if assumption*(D)*is satisﬁed,then u*> *in*.

In Section , we will prove the main results. Some corollaries and examples are shown in Section .

**2 The proof of the main results**

First, we present the main tool in this paper - the generalized concentration-compactness principle, which is easily deduced from Theorem . in [].

**Lemma .** *Suppose assumptions*(A), (A)*and*(A)*hold,unweakly converges to u in*

*X and* *μn*=*A(∇un*)dx,*νn*=|u*n*|*p*

*

*dx converge toμ*, *νweakly in the sense of measures,*
*respectively.*

*Then there exist some at most countable set J,a family*{x*j*;*j*∈*J}of distinct points in*,

*and two families*{*νj*;*j*∈*J},*{*μj*;*j*∈*J}of positive numbers such that*

*ν*=|u|*p***dx*+

*j∈J*

*νjδxj*, *μ*≥*A(∇u)dx*+

*j∈J*
*μjδxj*,

*μj*≥*S(νj*)*p*/*p*

*

(∀*j*∈*J),*

(.)

*whereδxjdenotes the Dirac measure at the point xj*.

Second, we deduce some properties of*a*and*f* by the similar method as in [] or [].
According to assumptions (A)-(A) and (B)-(B), we conclude that for any*σ*> ,*p*≤
*k*≤*p**_{, there exist some constants}_{C}

*k*,*σ*,*Cσ*,*C*such that for any*ξ*∈**R***n*,*x*∈,*u*∈**R**,

*pA(ξ*) –*a*|*ξ*|*p*+*a(ξ*)·*ξ*–*a*|*ξ*|*p*≤*σ*|*ξ*|*p*+*Cσ*, (.)

*c*
p*|u|

*p**_{–}_{C}_{≤}_{F(x,}_{u)}_{≤}*b(x) +σ*

*p* |u|

*p*_{+}*c*+*σ*
*p** |u|

*p**_{+}_{C}

*k*,*σ*|u|*k*, (.)

*c*
|u|

*p**_{–}_{C|u| ≤}_{f}_{(x,}_{u)u}_{≤}_{b(x) +}_{σ}_{|u|}*p*_{+ (c}

+*σ*)|u|*p*

*

+*Ck*,*σ*|u|*k*, (.)

*f*(x,*u)u*–*p***F(x,u)*≥–*σ*|*u*|*p**–*Cσ*,

+*σ*
*c*

*f*(x,*u)u*–*p***F(x,u)*≥–C*σ*.

(.)

To obtain a nonnegative solution, we ﬁrst consider the following variational functional and its Fréchet derivation:

*I(u) =*

*A(*∇*u)dx*–

*Fx,u*+*dx;* (.)

*I*(u),*υ*=

*a(∇u)*· ∇*υdx*–

*fx,u*+*υdx,* ∀*υ*∈*X.* (.)

**Lemma .***I() = and there exist two constantsα*,*ρ**and a function u*∈*X such that*

_{I(u)|}_{∂}_{B}_{ρ}

()≥*α*> , u *X*>*ρ*, *I(u*)≤, (.)

*where Bρ*() ={*u*∈*X,* *u* *X*≤*ρ*},*∂Bρ*()*denotes the boundary of Bρ*().

*Proof* Let*u*= in (.), and we have*I() = .*
Choosing*k*=*p**_{,}_{σ}_{=}_{a}

*βS/ in (.) (whereS*=inf*υ*∈X\{} *υ* *pX* *υ*

–*p*

*p** is the best

embed-ding constant from*X*to*Lp**()), and combining assumptions (C), (A), we have

*I(u)*≥

*A(*∇*u)dx*–

b(x) +*a**βS*

p *u*

+*p*

+*Cu*+*p***dx*≥*a**β*
p *u*

*p*
*X*–*C* *u*

*p**
*X*.

Hence,*I(u) > if* u *X*is small enough. So, we have showed the existence of*α*and*ρ*in
(.).

Next, we construct*u*satisfying (.). In fact, ﬁxing a nonnegative and nontrivial
func-tion*u*, recalling assumption (A) and (.), we deduce that there are positive constants
*C*,*C*,*C*such that

*I(tu*)≤

*a**tp*|∇*u*|*p*+*a**tm*|∇*u*|*m*

*dx*–

*c**tp*

*

p* *u*
+*p**_{+}_{C}

*dx*

≤*C**tp*–*C**tp*

*

+*C*.

Hence, if*t*is large enough, then we can set*u*=*tu*satisfying (.).

According to the Ambrosetti-Rabinowitz mountain pass theorem without the (PS)
con-dition, there exists a function sequence{u*n*}∞*n*=⊂*X*such that as*n*→ ∞,

*I(un*) –→*c* and *I*(u*n*) –→ in*X**, where*c*=inf

*γ*∈max*u∈γI(u)*≥*α*> , (.)

denotes the class of continuous paths joining to*u* in*X,X**denotes the dual space
of*X.*

**Lemma .** *The sequence*{*un*}∞*n*=*is bounded in X.*

*Proof* Let*υ*=*un*in (.) and combine (.), (.). We see that as*n*→ ∞,

*c*+*o() =I(un*) –*δI*(u*n*),*un*

= ( –*pδ*)

*A(*∇*un*)*dx*

+*δ*

*pA(∇un*) –*a(∇un*)· ∇u*n*

+

*fx,u*+* _{n}un*–

*δF*

*x,u*+_{n}dx

≥*a*( –*pδ*) – *δσ*

u*n* *pX*–*Cσ*, where*δ*=

*p**

+*σ*
*c*

.

In the last inequality, we have used (.) and (.). If we ﬁx a small enough*σ* such that
*a*( –*pδ*) – *δσ*> in the above inequality, then the conclusion in this lemma is obvious.

As a result of the above preparations, we can prove Theorem ..

*Proof of Theorem*. Since{u*n*}∞*n*=is bounded in*X, it is easy to see there are au*∈*X*and
a subsequence of{u*n*}∞*n*=, still denoted by itself , such that

*unu* weakly in*X,*

*un*→*u* a.e. inand strongly in*Lq*()

≤*q*<*p**,

(.)

*fx,u*+* _{n}fx,u*+ weakly in

*X**. (.)

By the Helly theorem, there exist a subsequence, still denoted by itself, and two
nonneg-ative measures*μ*and*ν*such that as*n*→ ∞,

*A(∇un*)*dx*
*w*

–→*μ*, |u*n*|*p*

*

*dx*–→*w* *ν* weakly in the sense of measures. (.)

Applying Lemma ., we have the corresponding conclusions of Lemma ..
Next, we establish the lower-bound of*νj*and*μj*

*νj*≥(pS)*p*

*_{/(}* _{p}**

_{–}

_{p}_{)}

*cp*_{}*/(*p*–*p*)*> , *μj*≥*pp*/(*p*

*_{–}_{p}_{)}

*Sp**/(*p**–*p*)*c*_{}*p*/(*p*–*p**). (.)

Denote*ϕ*as the cutoﬀ function of the ball*B*() in**R***N*,*i.e., which satisﬁes*

*ϕ*∈*C*_{}∞**R***N*, ≤*ϕ*≤,

*ϕ*(x) = if*x*∈*B*(), *ϕ*(x) = if*x*∈*B*().

(.)

Deﬁne*ϕε*,*j*=*ϕ*((x–*xj*)/*ε*) for every*ε*> and*xj*∈**R***N* for any*j*∈*J. Recalling Lemma in*

[], we have the following estimate:

|u*n*∇*ϕε*,*j*|*pdx*≤*S*–
**R***N*|∇

*ϕ*|
*p***p*
*p**–*p _{dx}*

*p**–*p*
*p**

*B*(*xj*,*ε*)

|∇*un*|*pdx*

≤*C*

*B*(*xj*,*ε*)

|∇*un*|*pdx.* (.)

Hence,{u*nϕε*,*j*}∞*n*=is still bounded in*X*and the boundary is independent of*ε*,*j.*

Let*u*=*un*,*υ*=*unϕε*,*j*in (.) and combine (.), (.), (.), (.). Then we obtain as

*n*→ ∞,

*o() =I*(u*n*),*unϕε*,*j*

=

*a(∇un*)·(∇u*nϕε*,*j*+*un*∇*ϕε*,*j*) –*f*

*x,u*+* _{n}ϕε*,

*jundx*

≥*p*

*Bε*/(*xj*)

*A(*∇*un*)dx– *σ* *un* *pX*–*Cσ*mes

*Bε*()

–

*Bε*(*xj*)

(c+*σ*)|u*n*|*p*

*

+*Cσ*

*dx*

–

* _{a(∇}un*)

*p*

*p*–

_{dx}*p*–

*p*

*C*

*B*(*xj*,*ε*)

|∇*un*|*pdx*+*o()*

In the above inequality, ﬁrst letting*n*→ ∞, then taking*ε*→+_{, and ﬁnally taking}_{σ}_{→}_{}+_{,}
we deduce*pμj*≤*c**νj*. So, (.) implies (.). Since*ν*is a bounded measure,*J*is at most

ﬁnite.

In the following, we prove that*u*≥, and it is the solution of problem (.). If*J*is empty,
then the proof is similar to that when*J*is nonempty, which we omit. Next, we suppose
*J* is nonempty and denote*J*={, , . . . ,*m},ε*={x∈|dist(x,*xj*) >*ε*,∀j∈*J}. Fix a large*

enough*R*and a suﬃciently small*ε*so that

¯

⊂*BR*(), *Bε*(x*i*)∩*Bε*(x*j*) =Ø while*i*=*j* and

*m*

*j*=

*Bε*(x*j*)⊂*BR*().

Deﬁne*ψε*(x) =*ϕ*(x/R) –

*m*

*j*=*ϕε*,*j*(x) with*x*∈**R***N*, <*ε*≤*ε*. It is not diﬃcult to deduce
that{*ψεun*}∞*n*=is bounded in*X*and the bound is independent of*ε*from (.). According
to (.) and (.), it is clear that as*n*→ ∞,

*o() =I*(u*n*) –*I*(u), (u*n*–*u)ψε*

=

*J*(u*n*,*u)ψε*+*J*(u*n*,*u) +J*(u*n*,*u)dx,* (.)

where*J*(u*n*,*u) = (a(*∇*un*) –*a(*∇*u))*·(∇*un*–∇*u),J*(u*n*,*u) = (a(*∇*un*) –*a(*∇*u))*· ∇*ψε*(u*n*–*u)*

and*J*(u*n*,*u) = (f*(x,*u*+) –*f*(x,*u*+*n*))*ψε*(u–*un*). Applying the method as in (.), we see that

|*J*(u*n*,*u)dx*|converges to as*n*→ ∞. Moreover, the deﬁnition of*ψε*and (.) imply

lim

*n→∞* * _{}*|u

*nψε*|

*p**

_{dx}_{=}

|*ψε*|*p*

*

*dν*=

|u*ψε*|*p*

*

*dx.*

Recalling (.), we see*unψε*→*uψε*in*Lp*

*

(). In view of (.), we obtain

* _{}J*(u

*n*,

*u)dx*

≤

_{f}

*x,u*++*fx,u*+_{n}

*p**
*p**–_{dx}

*p**–

*p** ψ

*ε*(u*n*–*u) _{p}**→. (.)

According to (.), (.), we deduce that*J*(u*n*,*u) converges to a.e. in*as*n*→ ∞,

maybe extracting a subsequence. Since *A(ξ*) is strictly convex, by the same method as
[], we claim∇u*n*→ ∇ua.e. in, and there exists a subsequence, still denoted by itself,

such that*a(∇un*) weakly converges to*a(∇u) inX**. Hence, (.), (.) and (.) imply that
*I*(u*n*) weakly converges to*I*(u) in*X**and*I*(u) = . So, it is not diﬃcult to see that*u*≥

and*I*(u) =*I*(u) = , which means that*u*is a weakly solution of equation (.).

Next, we prove*u*is nontrivial,*i.e.,* *u*= if assumption (C) holds. According to the
deﬁnition of*I(u) and the properties of*{u*n*}∞*n*=, we conclude that as*n*→ ∞,

*o() +c*–*I(u)*

–
*p**

*j∈J*

*I*(u*n*),*unϕε*,*j*

=

*A(∇un*) –*A(∇u)*

*dx*–

–
*p**

*j∈J*

*a(∇un*)· ∇(u*nϕε*,*j*) –*f*

*x,u*+* _{n}ϕε*,

*jundx*

=*K*+*K*+
*p**

*j∈J*

*p***K*,*j*+

*p**–*p*

*A(∇un*)*ϕε*,*jdx*+*K*,*j*–*K*,*j*+*K*,*j*

, (.)

where*K*,*j*=

(F(x,*u*

+_{) –}_{A(∇}_{u))}_{ϕ}

*ε*,*jdx*converges to as*ε*→+, and it has been proved

in (.) that*K*,*j*=

*a(∇un*)· ∇*ϕε*,*jundx*converges to as*n*→ ∞. Moreover, as*n*→ ∞,

*K*=

*A(∇un*) –*A(∇u)*

*ψεdx*

=

*ψεdμ*–

*ψεA(*∇*u)dx*+*o()*≥*o()*

by (.),

*K*=

*Fx,u*+* _{n}*–

*Fx,u*+

*ψεdx*=

*o()*

by the method similar to that in (.),

*K*,*j*=

*pA(*∇*un*) –*a(*∇*un*)· ∇*un*

*ϕε*,*jdx*

≥–*σ* *un* *pX*–*Cσ*mes

*Bε*() by (.)

,

*K*,*j*=

*fx,u*+* _{n}un*–

*p**

*F*

*x,u*+* _{n}ϕε*,

*jdx*

≥–*σ* *un* *p*

*

*p**–*Cσ*mes

*Bε*() by ( .)

.

Combining (.), (.) and the above inequalities and equalities, ﬁrstly taking*n*→ ∞,
then taking*ε*→+_{, and ﬁnally taking}_{σ}_{→}_{}+_{in (.), we have}

*c*–*I(u) =c*–*I(u)*≥
*p**–*p*

*p** *μj*≥

*p*–

*p**

(pS)

*p**
*p**–*p _{c}*

*p*
*p*–*p**

.

According to the deﬁnition of *c*and assumption (C), we have*I(u) < , which means*

*u*= .

Before proving Theorem ., we need to introduce the function family{*υε*}which

ap-proximates the best embedding constant*S*from*X*to*Lp**_{(}_{}_{). Without loss of }

generaliza-tion, we suppose*B*()⊂, <*ε*< . Denote

*U*=

( +|x|*p*/(*p*–)_{)}(*N*–*p*)/*p*, *Uε*=*ε*
*p*–*N*

*p* _{U}

*x*

*ε*

, *uε*=*Uεϕ*, *υε*=*uε* *uε* –* _{p}**,

where*ϕ*is deﬁned in (.), and*U*is the extremal function reaching*S. It is easy to check*
that as*ε*→+,

**R***N*_{\B}

()

|U*ε*|*p*

*

*dx*=*OεpN*–_{,}

**R***N*_{\B}

()

|∇U*ε*|*pdx*=*O*

*ε*

*N*–*p*

*p*–_{,} _{(.)}

*B*()\B()

|*Uε*|*mdx*=

*B*()\B()

|∇*Uε*|*mdx*=*O*

*ε*

(*N*–*p*)*m*

(*p*–)*p*_{,} _{(.)}

*υε* *p*

*

*p**= , *υε* *p _{p}*=

*o(),*

*υε*

*p*=

_{X}*S*+

*O*

As a result of the preparations, we can prove Theorem . as follows.

*Proof of Theorem*. Without loss of generalization, we suppose*B*()⊂*W*. For
conve-nience, let *A*and*B*denote some positive constants which may be diﬀerent in diﬀerent
places.

Applying the method in the proof of Lemma ., we deduce that*I(tυε*)→–∞as*t*→

+∞, which implies there exists a*tε*≥ such that*I(tευε*) =sup_{t≥}_{}*I(tυε*) and

*d*

*dtI(tυε*)|*t*=*tε*=

*I*(t*ευε*),*υε*

=

*a(tε*∇*υε*)· ∇*υεdx*–

*f*(x,*tευε*)*υεdx*= . (.)

Let*σ*= and*k*=*p*in (.), combining (.), (A), (A) and (A), we obtain

*a**tεp* *υε* *p _{X}*≤

*A(tε*∇*υε*)*dx*≤*tε*

*a(tε*∇*υε*)· ∇*υεdx*

≤*Ctp _{ε}*

*υε*

*pp*+

*tp*

*

*ε* *υε* *p*

*

*p**

.

Recalling (.), we see that*tε*is positive and bounded away from zero as*ε*→+.

More-over, according to assumption (A) and (.), we infer

*Ct _{ε}p*

*υε*

*pX*+

≥*tε*

*a(tε*∇*υε*)· ∇*υεdx*=

*f*(x,*tευε*)t*ευεdx*

≥*c*
*t*

*p**

*ε* *υε* *p*

*

*p**–*C.*

Hence, (.) implies*tε*is bounded, and the bound is independent of*ε*.

Set

*h*(t) =
*a*

*pt*

*p* _{υ}

*ε* *p _{X}*–

*c*
*p***t*

*p** _{υ}

*ε* *p*

*

*p**.

In view of (.), we have

max

*t≥**h*(t) =

*p*–
*p**

*a* *υε* *p _{X}*

*υε* *p _{p}**

*p**
*p**–*p*

*c*

*p*
*p*–*p**

=
*p*–
*p**

(a*S)*

*p**
*p**–*p _{c}*

*p*
*p*–*p**

+*O*

*ε*

*N*–*p*
*p*–_{.}

It is noted that*φ*(r) is increasing with*φ*() = according to assumption (C). Hence, we
deduce

sup

*t≥*

*I(tυε*) =*I(tευε*)≤*h*(t*ε*) +

*p*

*φtε*|∇*υε*|

*dx*–

*ψ*(t*ευε*)*dx*

≤
*p*–
*p**

(a*S)*

*p**
*p**–*p _{c}*

*p*
*p*–*p**

+*Oε*
*N*–*p*

*p*–_{+}

*p*

*φtε*|∇*υε*|

*dx*–

*ψ*(t*ευε*)*dx.* (.)

Next, we handle the last two terms on the right-hand side of (.). Since*φ*(r) is
nonneg-ative and increasing, we can utilize the properties of*υε*and*tε*to calculate

≤

*φtε*|∇*υε*|

*dx*≤

*B*()

*φA|∇Uε*|

*dx*+

*B*()\B()

*φA|∇uε*|

*B*()

*φA|∇Uε*|

*dx*≤

*B*/*ε*()

*εNφAε*–*pN*|∇U|_{dx}

≤*BεN*

*ε*–

+

*Aε*–
*ε*–

(r)*dr,* (.)

where(r) is deﬁned in assumption (C). Without loss of generalization, suppose*A*>
and <*ε*< . According to the deﬁnition of*φ* and assumption (A), it is clear that ≤

*φ*(r)≤*a**rp*+*a**rm*, then we have

*BεN*
*Aε*–
*ε*–

(r)*dr*=

*B*()\B()

*φA*|∇*uε*|

*dx*=*Oε*

(*N*–*p*)*m*

(*p*–)*p*_{.} _{(.)}

We have used (.) and (.) in the last equality. Furthermore, (.) and the deﬁnition
of*ψ*(u) imply that*ψ*(u)≤*B(*|*u*|*p*_{+}_{|}_{u}_{|}*p**_{). Repeating the above argument, we obtain}

*ψ*(t*ευε*)*dx*≥*BεN*
*ε*–

(r)*dr*+*Oε*
*N*–*p*

*p*–_{.} _{(.)}

Remembering assumption (A), we see*pS*≥*a**S. Combining (.)-(.), we obtain*

sup

*t≥*

*I(tυε*)≤

*p*–

*p**

(pS)

*p**
*p**–*p _{c}*

*p*
*p*–*p**

–*εN*

*BK*(*ψ*) –*BK*(*φ*)

+*Oε*

(*N*–*p*)*m*

(*p*–)*p*_{.}

So, assumption (C) implies that assumption (C) holds if we choose*ε*small enough, and

the conclusion in Theorem . follows from Theorem ..

*Proof of Theorem*. We ﬁrstly prove*u*∈*L*∞() by the Moser iteration. Since the problem
involves critical growth, we need some preparation before making the Moser iteration. Set

*η*(t)∈*C(***R**) and

*η*(t) =

⎧ ⎨ ⎩

sgn(t)|*t*|*k* _{if}_{|}_{t}_{| ≤}_{M,}

linear if|t| ≥*M,*

where*k*> ,*M*> , *ξ*(t) =

*t*

*η*(s)p*ds.*

It is not diﬃcult to check that*η*(t) > and*ξ*(u)∈*X*if*u*∈*X, and for anyt*∈**R**,

ξ(t)≤*k*η(t)η(t)p–, *η*(t)≤*k* +η(t),

ξ(t)|t|*p*–≤*kp*η(t)*p*.

(.)

Let*υ*(x) =*ξ*(u(x))*ψp*(x) in (.), where*ψ*(x) =*ϕ*((x–*x*)/R) and*ϕ*is deﬁned in (.),
*x*⊂,*R*> . Denote*D*=*B**R*(x)∩,*E*=*BR*(x)∩, then we compute

*a(∇u)*· ∇*uξ*(u)*ψpdx*+*p*

*a(∇u)*· ∇*ψ ξ*(u)*ψp*–*dx*

=

Denote the ﬁrst term and the second term on the left-hand side and on the right-hand side
of (.) as*J*and*J*,*J*, respectively. Now, we estimate them as follows:

*J*≥*a*

|∇*u*|*pψpξ*(u)dx=*a*

_{∇}*η*(u)*pψpdx*

=*a*

*D*
_{∇}

*η*(u)*ψp*–η(u)∇*ψpdx,*

*J*≥–pC

|∇*u*|*p*–+|∇*u*|*m*–*ξ*(u)|∇*ψ*|*ψp*–*dx*

≥–pkC

*D*

*η*(u)∇u*p*–+η(u)*p*–|∇*u|m*–*η*(u)|∇*ψ*|*ψp*–*dx* by (.)

≥–*σ*
*D*

_{∇}*η*(u)*pψpdx*–*σJ*–*Cσkp*

*D*

*η*(u)*p*|∇*ψ*|*pdx,*

where*σ* is a positive number deﬁned later and

*J*=

*D*

*η*(u)*p*|∇*u*|*mp*––*p _{ψ}p_{dx}*≤

_{k}p*D*

+*η*(u)*p* +|∇*u*|*pψpdx* by ( .)

≤

*D*

_{∇}*η*(u)*pψpdx*+*kp*

*D*

η(u)*pψpdx*+*Ckp*,

*J*≤*C*

|u|*p**–_{+ }_{ξ}_{(u)}_{ψ}p_{dx}

≤*Ckp*

|u|*p**–*p*η_{(u)}*p*

*ψp*+η(u) +η(u)*p*–*ψpdx* by (.)

≤*Ckp*

*D*

η(u)*ψp***dx*

*p*
*p**

*D*

|u|*p**_{dx}*p**–*p*

*p**

+mes(D)

*p**–*p*
*p**

+*Ckp*.

Set*k*=*k*=*p**/pin view of (.) and the above equalities. If we ﬁrstly ﬁx a small enough

*σ*, then a small enough*R*, then we can obtain

*E*

*η*(u)*p***dx*

p/*p**

≤*C*

*E*

_{∇}*η*(u)*pdx*≤*C*

*D*

*η*(u)*pdx*+*C,*

where*C*is a constant independent of*M. TakingM*→+∞, then we deduce*u*∈*Lk**p**_{(E).}

Applying a simple covering argument, we achieve that*u*∈*Lk**p**_{(}_{}_{). Finally, repeating the}

same argument, we derive that*u*∈*Lp**+*p*_{(}_{}_{). As a result of the preparations, we can use}

the Moser iteration to prove*u*∈*L*∞(). Let*υ*(x) =*ξ*(u(x)) in (.), and we calculate

*a*

_{∇}*η*(u)*pdx*≤

*a(*∇*u)*· ∇*uξ*(u)*dx*=

*f*(x,*u)ξ*(u)*dx*

≤*C*

|u|*p**–_{+ }_{ξ}_{(u)}_{dx}

≤*Ckp*

|*u*|*p**–*p*+ *η*(u)*p*+ *dx* by (.)

≤*Ckp*

*η*(u)(*p**+*p*)/*dx*

*p*
*p**+*p*

|*u*|*p**+*pdx*

*p**–*p*
*p**+*p*

+

where*C*is a constant independent of*M*and*k. If we setλ*= (p*_{+}* _{p)/(p}**

_{), then the above}inequality implies

*u* * _{kp}**≤

*C*/

*kpk*/

*k*

+ *u* *kp _{λ}_{kp}**

/(*kp*)
.

Thus,*u*∈*L*∞() follows from the standard Moser iteration method.

Applying Theorem in [], we see that there exists a constant *α*∈(, ) such that
*u*∈*C*,*α*_{(}_{}_{). If we rewrite}_{f}_{(x,}_{u) as (f}_{(x,}_{u)/u)u, then}_{f}_{(x,}_{u)/u}_{∈}* _{L}*∞

_{(}

_{}_{) with}

_{f}_{(x,}

_{u)/u}_{≥}

_{.}

Employing Theorem in [], it is obvious that*u*> in.

**3 Some corollaries and examples**

In this section, we ﬁrstly consider when (C) is true through analyzing*K*(*ψ*) and*K*(*φ*),
then we give some concrete examples and corollaries.

Firstly, we analyze the eﬀect of*a(z) toK*(*φ*):

**Lemma .** *Suppose a(z)satisﬁes assumptions*(A)-(A)*and*

(A) *There exist positive numbersm*,*m*,*A,Bsuch thatφ*(r)≤*Brm**for any*≤*r*≤*A*

*andφ*(r)≤*Brm* * _{for any}_{r}*≥

_{A.}*Then K*(*φ*) =*O(λ*(*ε*) +*λ*(*ε*))*asε*→+,*where K*(*φ*)*is deﬁned in Theorem*.*and*

*λ*(*ε*) =

⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
*ε*

(*N*–*p*)*m*_{}

(*p*–)*p* –*N* _{if m}

<*N _{N}*(

*p*

_{–}–),

*ε*

–*m*_{}*N*

*p* |_{ln}* _{ε}*|

_{if m}=*N _{N}*(

*p*

_{–}–),

*ε*
*N*(–*p*)

(*N*–)*p* _{if m}

>*N _{N}*(

*p*

_{–}–),

*λ*(*ε*) =

⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
*ε*

*N*(–*p*)

(*N*–)*p* _{if m}

<*N _{N}*(

*p*

_{–}–),

*ε*

–*m**N*

*p* |_{ln}* _{ε}*|

_{if m}=*N _{N}*(

*p*

_{–}–),

*ε*

–*m*_{}*N*

*p* _{if m}

>*N _{N}*(

*p*

_{–}–).

*Proof* Repeating the argument similar to (.), we compute

≤*K*(*φ*)≤*B*

*Aε*(*p*–)*N*/*p*

*ε*

–*m**N*
*p* * _{r}N*+

*m*

*p*––_{dr}_{+}_{B}

*ε*–

*Aε*–

*N*(*p*–)
(*N*–)*p*
*ε*

–*m**N*

*p* * _{r}N*–

*Np*––

*m*–

*dr*

+*B*

*Aε*(*p*–)*N*/*p*

*ε*

–*m*_{}*N*
*p* * _{r}N*+

*m*_{}

*p*––_{dr}_{+}_{B}

*Aε*–

*N*(*p*–)
(*N*–)*p*

*ε*

–*m*_{}*N*

*p* * _{r}N*–

*Np*––

*m*–

_{dr}=*λ*(*ε*) +*λ*(*ε*).

Secondly, we analyze how*ψ*(u) aﬀects*K*(*ψ*). The proof is similar to the above, and we
omit it.

**Lemma .** *Supposeψ*(u)*deﬁned in Theorem*.*satisﬁes*

(B) *There are positive numbersA,Bandq*≥*psuch thatψ*(u)≥*B|u|q _{when}*

_{}

_{≤}

_{u}_{≤}

_{A.}*Then we have K*(*ψ*)≥*λ*(*ε*)*asε*→+,*where*

*λ*(*ε*) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*Cε*

(*N*–*p*)*q*

(*p*–)*p*–*N* _{if q}_{<}*N*(*p*–)
*N*–*p* ,

*CεN*(–*pp*)|_{ln}* _{ε}*|

_{if q}_{=}

*N*(

*p*–)

*N*–

*p*,

*Cε*
*N*(–*p*)

*p* _{if q}_{>}*N*(*p*–)
*N*–*p* ,

*C is a positive constant.*

(B) *There exist positive numbersA,Bandq*<*p**_{such that}_{ψ}_{(u)}_{≥}_{B|u|}q_{if}_{u}_{≥}_{A.}

*Then K*(*ψ*)≥*λ*(*ε*)*asε*→+,*where*

*λ*(*ε*) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*CεN*(–*pp*) _{if q}_{<}*N*(*p*–)
*N*–*p* ,

*Cε*
*N*(–*p*)

*p* |_{ln}* _{ε}*|

_{if q}_{=}

*N*(

*p*–)

*N*–

*p*,

*Cε*

(*p*–*N*)*q*

*p* _{if q}_{>}*N*(*p*–)
*N*–*p* ,

*C is a positive constant.*

In the following, we can utilize Theorem . and Lemmas .-. to prove the following results about some concrete problems. The proof is trivial and we omit it.

**Corollary .** *Assume a(ξ*) = ( +|*ξ*|)*p*– _{ξ}_{,}* _{p}*≥

_{}

_{in problem}_{(.),}

_{and assumptions}_{}

(B)-(B), (C)*hold,ψ*(r)*deﬁned in assumption*(B)*satisﬁes*

(B) lim

*ε*→+

*K*(*ψ*)

*λ*(*ε*)

= +∞, *whereλ*(*ε*) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

*ε*–*N*+

(*N*–*p*)

*p*(*p*–) _{if p}_{> –}

*N*,

*ε*–*N*+

(*N*–*p*)

*p*(*p*–)|_{ln}* _{ε}*|

_{if p}_{= –}

*N*,

*ε*–
*N*(*p*–)

(*N*–)*p* _{if p}_{< –}
*N*.

*Then problem*(.)*possesses a nontrivial solution.*

*Proof* Take*m*= and*m*=*p*– in Lemma ., and we can deduce the conclusion.

**Example .** Next, we consider the following equation:

⎧ ⎨ ⎩

–div(( +|∇*u|*_{)}*p*– ∇u) =*c|u|p**–_{u}_{+}*k(x)|u|q*–_{u}_{+}_{g(x,}_{u),}* _{x}*∈

_{}_{,}

*u*= , *x*∈*∂*. (.)

**Corollary .** *Suppose the parameters*≤*p*<*q*<*p**_{and c}_{> ,}_{the functions k(x)}_{∈}_{C(}_{}_{)}

*with k(x)*≥*k**_{> ,}_{and g}_{(x,}_{u)}_{∈}_{C(}_{}_{×}_{R}_{)}_{with g(x, ) = ,}_{g(x,}_{u)u}_{≥}_{.}_{Moreover,}

*q*>

⎧ ⎨ ⎩

*p**_{–} *N*

*N*– *if*≤*p*≤ –

*N*,

*p**_{–}

*p*– *if p*≥ –

*N*,

lim

*u→∞*sup_{x∈}_{}

|g(x,*u)|*
|u|*p**_{–} = ,

lim sup

*u→*
sup

*x∈*

*g(x,u)*
|*u*|*p*–* _{u}*<

*S.*

*Then problem*(.)*possesses a positive solution in C*,*α*_{(}_{}_{) (}_{α}_{∈}_{(, )).}

**Example .** Next, we consider the following equation:

⎧ ⎨ ⎩

–div(|∇u|*p*–_{∇}_{u}_{+}|∇u|*α*+*p*–

+|∇u|*p* ∇u) =*c|u|p*

*_{–}

*u*+*k(x)|u|q*–_{u}_{+}_{g(x,}_{u),}_{x}_{∈}_{}_{,}

**Corollary .** *Suppose the parameters*≤*α*<*p and q*<*p**_{,}_{c}_{> ,}_{the functions k(x)}_{∈}
*C(*)*with k(x)*≥*k**_{> ,}_{and g}_{(x,}_{u)}_{∈}_{C(}_{}_{×}_{R}_{)}_{with g(x, ) = ,}_{g(x,}_{u)u}_{≥}_{.}_{Moreover,}

*q*>

⎧ ⎨ ⎩

*p**_{–} *N*

*N*– *ifα*≤

*N*(*p*–)
(*N*–),

*Nα*

*N*–*p* *ifα*≥
*N*(*p*–)

(*N*–),

lim

*u→∞*sup_{x∈}

|g(x,*u)|*

|u|*p**_{–} = , lim sup

*u→*
sup

*x∈*

*g(x,u)*
|u|*p*–* _{u}*<

*S.*

*Then problem*(.)*possesses a positive solution in C*,*α*_{(}_{}_{) (}_{α}_{∈}_{(, )).}

*Proof* Letting*m*= and*m*=*α*in Lemma ., and combining Lemma ., Lemma .

and Theorem ., we can derive the conclusion.

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

ZY brought out the problem and gave the proof of the existence of a nontrivial solution. GD suggested that the generalized C-C principle could be applied to this problem and proved the non-trivial solution could be a positive solution. HY improved the regularity of the solution and checked all of the proof.

**Author details**

1_{School of Mathematics Science, South China Normal University, Guangzhou, 510631, China.}2_{School of Mathematics}
and Computer Science, Guangdong University of Business Studies, Guangzhou, 510320, China.

**Acknowledgements**

The project is supported by National Natural Science Foundation of China (No. 11271143, 10901060, 10971073), the Natural Science Foundation of Zhejiang Province (No. Y6110775, Y6110789).

Received: 14 May 2012 Accepted: 13 September 2012 Published: 2 October 2012
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